Ellipse

Just like a circle has one focal point – center, ellipse has two. We are using those two foci and one positive number to define it… But, lets start from the beginning.

Through history people were always interested in the universe, especially planet movement. In the very beginnings of astronomy, it was considered that the Earth was in the center of the universe and that all planets revolve around it in concentric circles. Later on, when new technology was introduced, telescopes, better lenses and more collected data people started noticing some irregularities.

These irregularities were explained later on, when a man named Tycho Brahe set a theory where planets revolve around the Sun in elliptic paths. People used to think that those elliptic paths are nothing but irregular circles, but studying their properties they found out that ellipses were everything but irregular.

If are two fixed points of the plain M, and a positive real number greater than , ellipse is the set that contains all points of the plain for which the sum of the distances from points and is a constant and equal to .

To put it in a mathematical form, the definition states:

Let’s explain this definition using the drawing of the ellipse.

This will be valid for any point on the ellipse.

Points and are called foci. The midpoint of the segment is called the center of an ellipse.

The line cuts the ellipse in two points – A and B. Those points are called vertexes. Just the same, the center line of the segment cuts the ellipse in points C and D. Those points are also called vertexes.

Segment is called the major, and segment is called the minor axis.

Segments and are called semi-major axes whose length is usually marked with . Segments and are called semi-minor axes whose length is usually marked with .

Since the two most important points in the ellipse are the foci, their distance from the center is also very important. Number is called linear eccentricity of the ellipse.

From here we can see an link between major axis, minor axis and linear eccentricity: